Sharp pathwise nonuniqueness for additive SDEs

Abstract

We construct a family of velocity fields demonstrating the sharpness of the classical Zvonkin--Veretennikov--Davie strong well-posedness by noise regime. We consider stochastic differential equations driven by Brownian noise with drift $u$ and show that for any $\hat{I}\pm <0$, there exists a velocity field $u\in L_t^{\infty }{C}^{\hat{I}}{\pm }_x$ that admits a unique weak solution but does not satisfy pathwise uniqueness (and hence has no strong solutions). This contrasts with the case $\hat{I}\pm \ge 0$, for which the existence of a unique strong solution is guaranteed. The velocity field construction is random, and the proof essentially uses central limit theorem scaling through the Berry--Esseen theorem. We also give natural extensions to non-Brownian driving noises, including nonuniqueness for arbitrary driving noises with certain Hölder regularities and an analogous sharpness of the strong well-posedness by noise regime for fractional Brownian motions.

33 pages